Volume 30, Issue 5

The effective sufficient conditions are established for the numerical approximation of solutions of the initial problem with weight for linear systems of ordinary differential equations with singularities.

Let 𝔄 {\mathfrak{A}} be a standard operator algebra on a complex Banach space 𝔛 {\mathfrak{X}} , dim ⁡ 𝔛 > 1 {\dim\mathfrak{X}>1} , and p n ⁢ ( T 1 , T 2 , … , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} the ( n - 1 ) {(n-1)} th-commutator of elements T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . Then every map ξ : 𝔄 → 𝔄 {\xi:\mathfrak{A}\rightarrow\mathfrak{A}} (not necessarily linear) satisfying ξ ⁢ ( p n ⁢ ( T 1 , T 2 , … , T n ) ) = ∑ i = 1 n p n ⁢ ( T 1 , T 2 , … , T i - 1 , ξ ⁢ ( T i ) , T i + 1 , … , T n ) {\xi(p_{n}(T_{1},T_{2},\dots,T_{n}))=\sum_{i=1}^{n}p_{n}(T_{1},T_{2},\dots,T_{% i-1},\xi(T_{i}),T_{i+1},\dots,T_{n})} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} is of the form ξ = Ω + Γ {\xi=\Omega+\Gamma} , where Ω : 𝔄 → 𝔄 {\Omega:\mathfrak{A}\rightarrow\mathfrak{A}} is an additive derivation and Γ : 𝔄 → ℂ ⁢ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} is a map that vanishes at each ( n - 1 ) {(n-1)} th-commutator p n ⁢ ( T 1 , T 2 , … , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 𝔄 {S\in\mathfrak{A}} and a linear map Γ : 𝔄 → ℂ ⁢ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} satisfying Γ ⁢ ( p n ⁢ ( T 1 , T 2 , … , T n ) ) = 0 {\Gamma(p_{n}(T_{1},T_{2},\dots,T_{n}))=0} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} , such that ξ ⁢ ( T ) = [ T , S ] + Γ ⁢ ( T ) {\xi(T)=[T,S]+\Gamma(T)} for all T ∈ 𝔄 {T\in\mathfrak{A}} .

Let A be a bounded linear operator defined on a complex Hilbert space and let | A | = ( A * ⁢ A ) 1 2 {|A|=(A^{*}A)^{\frac{1}{2}}} . Among other refinements of the well-known numerical radius inequality w 2 ⁢ ( A ) ≤ 1 2 ⁢ ∥ A * ⁢ A + A ⁢ A * ∥ {w^{2}(A)\leq\frac{1}{2}\|A^{*}A+AA^{*}\|} , we show that w 2 ⁢ ( A ) ≤ 1 4 ⁢ w 2 ⁢ ( | A ⁢ | + i | ⁢ A * | ) + 1 8 ⁢ ∥ | A | 2 + | A * | 2 ∥ + 1 4 ⁢ w ⁢ ( | A | ⁢ | A * | ) ≤ 1 2 ⁢ ∥ A * ⁢ A + A ⁢ A * ∥ . w^{2}(A)\leq\frac{1}{4}w^{2}(|A|+{\rm i}|A^{*}|)+\frac{1}{8}\||A|^{2}+|A^{*}|^% {2}\|+\frac{1}{4}w(|A||A^{*}|)\leq\frac{1}{2}\|A^{*}A+AA^{*}\|. Also, we develop inequalities involving the numerical radius and the spectral radius for the sum of the product operators, from which we derive the inequalities w p ⁢ ( A ) ≤ 1 2 ⁢ w ⁢ ( | A | p + i ⁢ | A * | p ) ≤ ∥ A ∥ p w^{p}(A)\leq\frac{1}{\sqrt{2}}w(|A|^{p}+{\rm i}|A^{*}|^{p})\leq\|A\|^{p} for all p ≥ 1 {p\geq 1} . Further, we derive new bounds for the zeros of complex polynomials.

The interval G = ( - 1 , 1 ) {G=(-1,1)} turns into a Lie group under the group operation x ∘ y := ( x + y ) ⁢ ( 1 + x ⁢ y ) - 1 {x\circ y:=(x+y)(1+xy)^{-1}} , x , y ∈ G {x,y\in G} . This enables us to define of the invariant measure d ⁢ G ⁢ ( x ) := ( 1 - x 2 ) - 1 ⁢ d ⁢ x {dG(x):=(1-x^{2})^{-1}dx} and the Fourier transformation ℱ G {{\mathcal{F}}{}_{G}} on the interval G and, as a consequence, we can consider Fourier convolution operators W G , a 0 := ℱ a G - 1 ℱ G {W^{0}_{G,a}:={\mathcal{F}}{}_{G}^{-1}a{\mathcal{F}}{}_{G}} on G . This class of convolutions includes the celebrated Prandtl, Tricomi and Lavrentjev–Bitsadze equations and, also, differential equations of arbitrary order with the generic differential operator 𝔇 G ⁢ u ⁢ ( x ) = ( 1 - x 2 ) ⁢ u ′ ⁢ ( x ) {\mathfrak{D}_{G}u(x)=(1-x^{2})u^{\prime}(x)} , x ∈ G {x\kern-1.0pt\in\kern-1.0ptG} . Equations are solved in the scale of generic Bessel potential ℍ p s ⁢ ( G , d ⁢ G ⁢ ( x ) ) {\mathbb{H}^{s}_{p}(G,dG(x))} , 1 ⩽ p ⩽ ∞ {1\kern-1.0pt\leqslant\kern-1.0ptp\kern-1.0pt\leqslant\kern-1.0pt\infty} , and Hölder–Zygmund ℤ ν ⁢ ( G ) {\mathbb{Z}^{\nu}(G)} , 0 < μ , ν < ∞ {0<\mu,\nu<\infty} , spaces, adapted to the group G . The boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a ⁢ ( ξ ) {a(\xi)} , ξ ∈ ℝ {\xi\in\mathbb{R}} , of a convolution equation W G , a 0 ⁢ u = f {W^{0}_{G,a}u=f} defines solvability as follows: the equation is uniquely solvable if and only if the symbol a is elliptic. The solution is written explicitly with the help of the inverse symbol. Also, we shortly touch upon the multidimensional analogue – the Lie group G n {G^{n}} .

The authors consider a fourth-order boundary value problem with a p -Laplacian appearing in both the equation and the boundary conditions. Upper and lower estimates for positive solutions to the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions to the problem are obtained and illustrated with an example.

In this paper, we mainly consider the singular k -Hessian equations S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) + g ⁢ ( | D ⁢ u | )   in  ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)+g(|Du|)\quad\text{in }\Omega and S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) ⁢ ( 1 + g ⁢ ( | D ⁢ u | ) )   in  ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))\quad\text{in }\Omega with the Dirichlet boundary condition u = 0 {u=0} on ∂ ⁡ Ω {\partial\Omega} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h , f and g . This problem is more difficult to solve than the k -Hessian problem without gradient terms, and requires additional new conditions in the proof process.

The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. As consequences, and apart from proving the other results, many known theorems can be either generalized or deduced.

We prove that a rational trigonometric series with its coefficients c ⁢ ( n ) = o ⁢ ( 1 ) {c(n)=o(1)} satisfying the condition ∑ n ∈ ℤ | Δ m ⁢ c ⁢ ( n ) | < ∞ {\sum_{n\in\mathbb{Z}}|\Delta^{m}c(n)|<\infty} , m ∈ ℕ {m\in\mathbb{N}} , converges pointwise to some f ⁢ ( x ) {f(x)} for every x ∈ ( 0 , 2 ⁢ π ) {x\in(0,2\pi)} and also converges in L p ⁢ [ 0 , 2 ⁢ π ) {L^{p}[0,2\pi)} -metric to f for 0 < p < 1 m {0<p<\frac{1}{m}} . This result is further extended to a double rational trigonometric series.

This paper deals with the introduction and study of a new, Orlicz type space associated with the Gegenbauer differential operator and briefly called here a G -Orlicz space. We prove that such spaces are Banach function spaces. The necessary and sufficient conditions are established which ensure the validity of weak and strong type inequalities for fractional maximal functions and fractional integrals of G -Orlicz spaces.

Let F be a free Leibniz algebra generated by the set X = { x 1 , … , x n } {X=\{x_{1},\dots,x_{n}\}} over the field K of characteristic 0 and let R be an ideal of F . In this study, a necessary and sufficient condition for n elements of the Leibniz algebra F / R ′ {F/R^{\prime}} to be a generating set is given.

Based on the uncertainty theory, Liu [B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 2009, 1, 3–10] introduced an uncertain integral for applying uncertain differential equation, finance, control, filtering and dynamical systems. Since uncertain integrals are the important content of uncertainty theory, this paper explores an approach of the relationship between uncertain integrals by the well-known Chebyshev-type inequality. Also, we propose the concept of an uncertain fractional integral which is generalized version of an uncertain integral. The definition of a strong comonotonic uncertain process and some new properties of the uncertain integral were presented in [C. You and N. Xiang, Some properties of uncertain integral, Iran. J. Fuzzy Syst. 15 2018, 2, 133–142]. Based on the strong comonotonic uncertain process, as an application, we provide Chebyshev’s inequality for a fractional uncertain integral and an uncertain integral.

In [S. Yang and Y. Lin, Multilinear strongly singular integral operators with generalized kernels and applications, AIMS Math. 6 2021, 12, 13533–13551], the authors of the present paper further weaken the smoothness condition of kernel functions with multilinear strongly singular Calderón–Zygmund operators of [Y. Lin, Multilinear theory of strongly singular Calderón–Zygmund operators and applications, Nonlinear Anal. 192 2020, Article ID 111699]. They defined a new class of multilinear strongly singular integral operators, and studied its weighted L p {L^{p}} boundedness, variable exponent L p ⁢ ( ⋅ ) {L^{p(\cdot)}} boundedness and endpoint estimates. In this paper, we naturally consider the boundedness of multilinear commutators and multilinear iterated commutators, which are generated by multilinear strongly singular integral operators with generalized kernels and Lipschitz functions. Our results include the corresponding results of multilinear strongly singular Calderón–Zygmund operators and classical multilinear Calderón–Zygmund operators, respectively.